Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,21,27,22,28,36,43,23,25,37,44,30,39,20,26)(19,29,33,40,34,45,31,46,32,47,35,24,38,41,48,42);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,30,34,23,32,22,38)(18,24,39,42,25,40,28,46)(20,35,37,48,36,33,21,31)(26,45,44,47,43,41,27,29);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(18,25)(20,36)(21,37)(24,40)(28,39)(31,48)(33,35)(42,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44419);